Mathematical modelling and symbolic dynamics analysis of three new Galton board models

A Galton board, also known as a quincunx, is a device invented by Francis Galton in 1873 that consists of two upright boards with rows of pins, and a funnel. In this paper, three new mathematical models of Galton board that are of increasing complexity are formulated. The discussion includes a brief literature review, the description of the systems, the important physical processes, the assumptions employed and the derivation of the governing equations of the models. The quincunx models are folded into a discrete-time deterministic dynamical system, called the quincunx maps, that enables a simplified analysis of the symbolic dynamics. While Galton and countless subsequent statisticians have suggested that a small ball falling through a quincunx would exhibit random walk; the results of the symbolic dynamics analysis demonstrate that this is not the case. This paper presents evidence that the details of the deterministic models are not essential for demonstrating deviations from the statistical models.